“best matching” et d´elais
Jean Dolbeault
http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine
12 Mai 2015
A. Fast diffusion equations: entropy, linearization, inequalities,
improvements
1
entropy methods
2
linearization of the entropy
3
improved Gagliardo-Nirenberg inequalities
Fast diffusion equations: entropy
methods
Existence, classical results
u t = ∆u m x ∈ R d , t > 0
Self-similar (Barenblatt) function: U (t) = O(t −d/(2 −d(1 −m)) ) as t → + ∞
[Friedmann, Kamin, 1980] k u(t, · ) − U (t, · ) k L
∞= o(t −d/(2 −d(1 −m)) )
d − 1 d
m fast diffusion equation
porous media equation heat equation
1
d − 2 d
global existence in L 1 extinction in finite time
Existence theory, critical values of the parameter m
Time-dependent rescaling, Free energy
Time-dependent rescaling: Take u(τ, y) = R −d (τ) v (t, y /R(τ)) where
dR
dτ = R d(1 −m) − 1 , R(0) = 1 , t = log R The function v solves a Fokker-Planck type equation
∂v
∂t = ∆v m + ∇ · (x v) , v | τ=0 = u 0
[Ralston, Newman, 1984] Lyapunov functional:
Generalized entropy or Free energy F [v ] :=
Z
R
dv m m − 1 + 1
2 | x | 2 v
dx − F 0
Entropy production is measured by the Generalized Fisher information
d
dt F [v ] = −I [v ] , I [v ] :=
Z
R
dv ∇ v m
v + x
2
dx
Relative entropy and entropy production
Stationary solution: choose C such that k v ∞ k L
1= k u k L
1= M > 0 v ∞ (x ) := C + 1 2m −m | x | 2 − 1/(1 −m)
+
Relative entropy: Fix F 0 so that F [v ∞ ] = 0 Entropy – entropy production inequality
Theorem
d ≥ 3, m ∈ [ d − d 1 , + ∞ ), m > 1 2 , m 6 = 1 I [v] ≥ 2 F [v ]
Corollary
A solution v with initial data u 0 ∈ L 1 + ( R d ) such that | x | 2 u 0 ∈ L 1 ( R d ),
u 0 m ∈ L 1 ( R d ) satisfies F [v(t , · )] ≤ F [u 0 ] e − 2 t
An equivalent formulation: Gagliardo-Nirenberg inequalities
F [v] = Z
R
dv m m − 1 + 1
2 | x | 2 v
dx −F 0 ≤ 1 2 Z
R
dv ∇ v m
v + x
2
dx = 1 2 I [v]
Rewrite it with p = 2m− 1 1 , v = w 2p , v m = w p+1 as 1
2 2m
2m − 1 2 Z
R
d|∇ w | 2 dx + 1
1 − m − d Z
R
d| w | 1+p dx − K ≥ 0 for some γ, K = K 0 R
R
dv dx = R
R
dw 2p dx γ
w = w ∞ = v ∞ 1/2p is optimal Theorem
[Del Pino, J.D.] With 1 < p ≤ d − d 2 (fast diffusion case) and d ≥ 3 k w k L
2p( R
d) ≤ C p,d GN k∇ w k θ L
2( R
d) k w k 1 L −
p+1θ ( R
d)
C p,d GN =
y(p− 1)
22πd
θ22y − d 2y
2p1Γ(y)
Γ(y −
d2)
θd, θ = p(d+2 d(p− − (d 1) − 2)p) , y = p+1 p − 1
... a proof by the Bakry-Emery method
Consider the generalized Fisher information I [v ] :=
Z
R
dv | z | 2 dx with z := ∇ v m v + x and compute
d
dt I [v (t, · )]+2 I [v (t , · )] = − 2 (m − 1) Z
R
du m (divz ) 2 dx − 2 X d i, j=1
Z
R
du m (∂ i z j ) 2 dx
the Fisher information decays exponentially:
I [v(t, · )] ≤ I [u 0 ] e − 2t lim t →∞ I [v(t, · )] = 0 and lim t →∞ F [v (t , · )] = 0
d dt
I [v (t, · )] − 2 F [v(t , · )]
≤ 0 means I [v ] ≥ 2 F [v]
[Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel,
Markowich, Toscani, Unterreiter], [Carrillo, V´ azquez]
The Bakry-Emery method: details (1/2)
With z (x, t) := η ∇ u m− 1 − 2 x, the equation can be rewritten as
∂u
∂t + ∇ · (u z ) = 0
(up to a time rescaling, which introduces a factor 2) and we have
∂z
∂t = η (1 − m) ∇ u m− 2 ∇ · (u z)
and ∇ ⊗ z = η ∇ ⊗ ∇ u m− 1 − 2 Id d
dt Z
R
du | z | 2 dx = Z
R
d∂u
∂t | z | 2 dx
| {z }
( I )
+ 2 Z
R
du z · ∂z
∂t dx
| {z }
( II )
(I) = Z
R
d∂u
∂t | z | 2 dx = Z
R
d∇ · (u z ) | z | 2 dx
= 2 η (1 − m) Z
R
du m− 2 ( ∇ u · z ) 2 dx + 2 η (1 − m) Z
R
du m− 1 ( ∇ u · z ) ( ∇ · z) dx + 2 η (1 − m)
Z
R
du m− 1 (z ⊗ ∇ u) : ( ∇ ⊗ z ) dx − 4 Z
R
du | z | 2 dx
The Bakry-Emery method: details (2/2)
(II) = 2 Z
R
du z · ∂z
∂t dx
= − 2 η (1 − m) Z
R
du m ( ∇ · z) 2 + 2 u m − 1 ( ∇ u · z ) ( ∇ · z ) + u m − 2 ( ∇ u · z ) 2 dx Z
R
d∂u
∂t | z | 2 dx + 4 Z
R
du | z | 2 dx
= − 2 η (1 − m) Z
R
du m− 2
u 2 ( ∇ · z) 2 + u ( ∇ u · z ) ( ∇ · z )
= − 2 η 1 − m m
Z
R
du m
|∇ z | 2 − (1 − m) ( ∇ · z ) 2 dx By the arithmetic geometric inequality, we know that
|∇ z | 2 − (1 − m) ( ∇ · z ) 2 ≥ 0
if 1 − m ≤ 1/d, that is, if m ≥ m 1 = 1 − 1/d
Fast diffusion: finite mass regime
Inequalities...
d − 1 d
m 1
d − 2 d
global existence in L 1
Bakry-Emery method (relative entropy) v m ∈ L 1 , x
2∈ L 1 Sobolev
Gagliardo-Nirenberg logarithmic Sobolev
d d +2
v
... existence of solutions of u t = ∆u m
Fast diffusion equations: the infinite mass regime by
linearization of the entropy
Extension to the infinite mass regime, finite time vanishing
If m > m c := d − d 2 ≤ m < m 1 , solutions globally exist in L 1 ( R d ) and the Barenblatt self-similar solution has finite mass
For m ≤ m c , the Barenblatt self-similar solution has infinite mass Extension to m ≤ m c ? Work in relative variables !
d−1 d
m 1
d−2 d
glob al existenc e i n L
1Bakry-Emer y m etho d ( relati ve en trop y) v
m∈ L
1, x
2∈ L
1d d+2
v v
0, V
D∈ L
1v
0− V
D∗
∈ L
1V
D1− V
D0∈ L
1[ V
D1V
D0] < ∞ [ V
D1V
D0] = ∞
m
1 d−4d−2
V
D1− V
D06∈ L
1m
cm
∗Gagliardo-Nire nb er g F
F
Entropy methods and linearization: intermediate asymptotics, vanishing
[A. Blanchet, M. Bonforte, J.D., G. Grillo, J.L. V´ azquez]
∂u
∂τ = −∇ · (u ∇ u m− 1 ) = 1 − m
m ∆u m (1)
m c < m < 1, T = + ∞ : intermediate asymptotics, τ → + ∞ R(τ) := (T + τ)
d(m−1mc)0 < m < m c , T < + ∞ : vanishing in finite time lim τ րT u(τ, y) = 0 R(τ) := (T − τ) −
d(mc1−m)Self-similar Barenblatt type solutions exists for any m t := 1 − 2 m log
R (τ) R(0)
and x := q
1 2 d | m − m
c|
y
R(τ)
Generalized Barenblatt profiles: V D (x) := D + | x | 2
m−11Sharp rates of convergence
Assumptions on the initial datum v 0
(H1) V D
0≤ v 0 ≤ V D
1for some D 0 > D 1 > 0
(H2) if d ≥ 3 and m ≤ m ∗ , (v 0 − V D ) is integrable for a suitable D ∈ [D 1 , D 0 ]
Theorem
[Blanchet, Bonforte, J.D., Grillo, V´ azquez] Under Assumptions (H1)-(H2), if m < 1 and m 6 = m ∗ := d− d− 4 2 , the entropy decays according to
F [v (t, · )] ≤ C e − 2 (1 − m) Λ
α,dt ∀ t ≥ 0
where Λ α,d > 0 is the best constant in the Hardy–Poincar´e inequality Λ α,d
Z
R
d| f | 2 dµ α − 1 ≤ Z
R
d|∇ f | 2 dµ α ∀ f ∈ H 1 (d µ α )
with α := 1/(m − 1) < 0, dµ α := h α dx, h α (x) := (1 + | x | 2 ) α
Plots (d = 5)
λ01=−4α−2d
λ10=−2α λ11=−6α−2 (d+ 2) λ02=−8α−4 (d+ 2)
λ20=−4α λ30 λ21λ
12 λ03
λcontα,d:=14(d+ 2α−2)2
α=−d α=−(d+ 2)
α=−d+2 2
α=−d−22 α=−d+62
α 0 Essential spectrum ofLα,d
α=−√d−1−d2 α=−√d−1−d+42 α=− −d+22
√2d (d= 5) Spectrum ofLα,d
mc=d−2d
m1=d−1d m2=d+1d+2
e m1=d+2d
e m2=d+4d+6
m Spectrum of (1−m)L1/(m−1),d
(d= 5)
Essential spectrum of (1
1
−m)L1/(m−1),d
2 4 6
Improved asymptotic rates
[Bonforte, J.D., Grillo, V´ azquez] Assume that m ∈ (m 1 , 1), d ≥ 3.
Under Assumption (H1), if v is a solution of the fast diffusion equation with initial datum v 0 such that R
R
dx v 0 dx = 0, then the asymptotic convergence holds with an improved rate corresponding to the improved spectral gap.
m1=d+2d
m1=d−1 d
m2=d+4d+6 m2=d+1d+2
4
2
m 1 mc=d−2
d (d= 5)
γ(m)
0
Higher order matching asymptotics
[J.D., G. Toscani] For some m ∈ (m c , 1) with m c := (d − 2)/d, we consider on R d the fast diffusion equation
∂u
∂τ + ∇ · u ∇ u m − 1
= 0
Without choosing R, we may define the function v such that u(τ, y + x 0 ) = R − d v (t, x) , R = R(τ) , t = 1 2 log R , x = y
R Then v has to be a solution of
∂v
∂t + ∇ · h v
σ
d2(m−m
c) ∇ v m− 1 − 2 x i
= 0 t > 0 , x ∈ R d with (as long as we make no assumption on R)
2 σ −
d2(m−m
c) = R 1 −d (1 −m) dR
dτ
Refined relative entropy
Consider the family of the Barenblatt profiles B σ (x) := σ −
d2C M + 1 σ | x | 2
m−11∀ x ∈ R d (2) Note that σ is a function of t : as long as dσ dt 6 = 0, the Barenblatt profile B σ is not a solution (it plays the role of a local Gibbs state) but we may still consider the relative entropy
F σ [v] := 1 m − 1
Z
R
dv m − B σ m − m B σ m − 1 (v − B σ ) dx The time derivative of this relative entropy is
d
dt F σ(t) [v(t , · )] = dσ dt
d dσ F σ [v]
| σ=σ(t)
| {z }
choose it = 0
⇐⇒ Minimize F σ [v ] w.r.t. σ ⇐⇒ R
R
d| x | 2 B σ dx = R
R
d| x | 2 v dx
+ m
m − 1 Z
R
dv m − 1 − B σ(t) m− 1 ∂v
∂t dx
The entropy / entropy production estimate
Using the new change of variables, we know that d
dt F σ(t) [v (t , · )] = − m σ(t)
d2(m−m
c) 1 − m
Z
R
dv ∇ h
v m − 1 − B σ(t) m− 1 i
2
dx
Let w := v/B σ and observe that the relative entropy can be written as F σ [v] = m
1 − m Z
R
dh w − 1 − 1
m w m − 1 i B σ m dx (Repeating) define the relative Fisher information by
I σ [v] :=
Z
R
d1
m − 1 ∇
(w m− 1 − 1) B σ m− 1
2
B σ w dx
so that d
dt F σ(t) [v (t , · )] = − m (1 − m) σ(t ) I σ(t) [v (t , · )] ∀ t > 0
When linearizing, one more mode is killed and σ(t) scales out
Improved rates of convergence
Theorem (J.D., G. Toscani)
Let m ∈ ( m e 1 , 1), d ≥ 2, v 0 ∈ L 1 + ( R d ) such that v 0 m , | y | 2 v 0 ∈ L 1 ( R d ) F [v(t, · )] ≤ C e − 2 γ(m) t ∀ t ≥ 0
where
γ(m) =
((d − 2) m − (d − 4))
24 (1 −m) if m ∈ ( m e 1 , m e 2 ] 4 (d + 2) m − 4 d if m ∈ [ m e 2 , m 2 ]
4 if m ∈ [m 2 , 1)
Spectral gaps and best constants
m
c=
d−2d0
m
1=
d−1dm
2=
d+1d+2e m
2:=
d+4d+6m 1
2 4
Case 1 Case 2 Case 3
γ(m)
(d= 5)
e
m
1:=
d+2dComments
1
A result by [Denzler, Koch, McCann]Higher order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach
2
The constant C in
F [v(t, · )] ≤ C e − 2 γ(m) t ∀ t ≥ 0
can be made explicit, under additional restrictions on the initial
data [Bonforte, J.D., Grillo, V´ azquez]
An explicit constant C ?
d
dt F [w (t, · )] = − I [w (t , · )] ∀ t > 0 h m − 2
Z
R
d| f | 2 V D 2 − m dx ≤ 2 F [w ] ≤ h 2 − m Z
R
d| f | 2 V D 2 − m dx where f := (w − 1) V D m − 1 , h := max { sup R
dw (t, · ), 1/inf R
dw (t , · ) } Z
R
d|∇ f | 2 V D dx ≤ h 5 − 2m I [w ] + d (1 − m)
h 4(2 − m) − 1 Z
R
d| f | 2 V D 2 −m dx 0 ≤ h − 1 ≤ C F
1−m d+2−(d+1)m
Corollary
F [w (t, · )] ≤ G t, h(0), F [w (0, · )]
for any t ≥ 0, where dG
dt = − 2 Λ α,d − Y (h)
[1 + X(h)] h 2 − m G , h = 1 + C G
d+2−1−(d+1)mm, G (0) = F [w (0, · )]
Gagliardo-Nirenberg and Sobolev inequalities :
improvements
[J.D., G. Toscani]
Best matching Barenblatt profiles
(Repeating) Consider the fast diffusion equation
∂u
∂t + ∇ · h u
σ
d2(m − m
c) ∇ u m − 1 − 2 x i
= 0 t > 0 , x ∈ R d with a nonlocal, time-dependent diffusion coefficient
σ(t ) = 1 K M
Z
R
d| x | 2 u(x, t) dx , K M :=
Z
R
d| x | 2 B 1 (x) dx where
B λ (x) := λ −
d2C M + λ 1 | x | 2
m−11∀ x ∈ R d and define the relative entropy
F λ [u] := 1 m − 1
Z
R
du m − B λ m − m B λ m − 1 (u − B λ )
dx
Three ingredients for global improvements
1
inf λ>0 F λ [u(x , t )] = F σ(t) [u(x, t)] so that d
dt F σ(t) [u(x, t)] = −J σ(t) [u( · , t)]
where the relative Fisher information is J λ [u] := λ
d2(m − m
c) m
1 − m Z
R
du ∇ u m − 1 − ∇ B λ m− 1 2 dx
2
In the Bakry-Emery method, there is an additional (good) term 4
"
1 + 2 C m,d F σ(t) [u( · , t )]
M γ σ 0
d2(1 −m)
# d
dt F σ(t) [u( · , t )]
≥ d
dt J σ(t) [u( · , t)]
3
The Csisz´ ar-Kullback inequality is also improved F σ [u] ≥ m
8 R
R
dB 1 m dx C M 2 k u − B σ k 2 L
1( S
d)
improved decay for the relative entropy
0.0 0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.0
t 7→ f(t) e
4tf (0)
t (a)
(b) (c)
Figure: Upper bounds on the decay of the relative entropy: t 7→ f (t) e
4t/f (0) (a): estimate given by the entropy-entropy production method
(b): exact solution of a simplified equation
(c): numerical solution (found by a shooting method)
A Csisz´ar-Kullback(-Pinsker) inequality
Let m ∈ ( m e 1 , 1) with m e 1 = d+2 d and consider the relative entropy F σ [u] := 1
m − 1 Z
R
du m − B σ m − m B σ m − 1 (u − B σ ) dx
Theorem
Let d ≥ 1, m ∈ ( m e 1 , 1) and assume that u is a nonnegative function in L 1 ( R d ) such that u m and x 7→ | x | 2 u are both integrable on R d . If k u k L
1( S
d) = M and R
R
d| x | 2 u dx = R
R
d| x | 2 B σ dx, then F σ [u]
σ
d2(1 − m) ≥ m 8 R
R
dB 1 m dx
C M k u − B σ k L
1( S
d) + 1 σ
Z
R
d| x | 2 | u − B σ | dx
2
Csisz´ar-Kullback(-Pinsker): proof (1/2)
Let v := u/B σ and dµ σ := B σ m dx Z
R
d(v − 1) dµ σ = Z
R
dB σ m− 1 (u − B σ ) dx
= σ
d2(1 −m) C M
Z
R
d(u − B σ ) dx + σ
d2(m
c−m) Z
R
d| x | 2 (u − B σ ) dx = 0 Z
R
d(v − 1) dµ σ = Z
v>1
(v − 1) dµ σ − Z
v<1
(1 − v ) dµ σ = 0 Z
R
d| v − 1 | dµ σ = Z
v>1
(v − 1) dµ σ + Z
v<1
(1 − v) dµ σ
Z
R
d| u − B σ | B σ m− 1 dx = Z
R
d| v − 1 | dµ σ = 2 Z
v<1 | v − 1 | dµ σ
Csisz´ar-Kullback(-Pinsker): proof (2/2)
A Taylor expansion shows that F σ [u] = 1
m − 1 Z
R
dv m − 1 − m (v − 1)
d µ σ = m 2
Z
R
dξ m− 2 | v − 1 | 2 dµ σ
≥ m 2
Z
v<1 | v − 1 | 2 dµ σ
Using the Cauchy-Schwarz inequality, we get R
v<1 | v − 1 | dµ σ 2
= R
v<1 | v − 1 | B σ
m2B σ
m2dx 2
≤ R
v <1 | v − 1 | 2 dµ σ R
R
dB σ m dx and finally obtain that
F σ [u] ≥ m 2
R
v<1 | v − 1 | dµ σ 2
R
R
dB σ m dx = m 8
R
R
d| u − B σ | B σ m − 1 dx 2
R
R
dB σ m dx
An improved Gagliardo-Nirenberg inequality: the setting
The inequality
k f k L
2p( S
d) ≤ C GN p,d k∇ f k θ L
2( S
d) k f k 1 L −
p+1θ ( S
d)
with θ = θ(p) := p− p 1 d+2 − d p (d − 2) , 1 < p ≤ d − d 2 if d ≥ 3 and
1 < p < ∞ if d = 2, can be rewritten, in a non-scale invariant form, as Z
R
d|∇ f | 2 dx + Z
R
d| f | p+1 dx ≥ K p,d
Z
R
d| f | 2 p dx γ
with γ = γ(p, d ) := d+2 d − −p p (d (d− − 4) 2) . Optimal function are given by f M,y,σ (x ) = 1
σ
d2C M + | x − y | 2 σ
−
p−11∀ x ∈ R d where C M is determined by R
R
df M,y 2p ,σ dx = M M d :=
f M,y,σ : (M, y, σ) ∈ M d := (0, ∞ ) × R d × (0, ∞ )
An improved Gagliardo-Nirenberg inequality (1/2)
Relative entropy functional R (p) [f ] := inf
g ∈ M
(p)d
Z
R
dh g 1 −p | f | 2p − g 2p
− p+1 2 p | f | p+1 − g p+1 i dx
Theorem
Let d ≥ 2, p > 1 and assume that p < d /(d − 2) if d ≥ 3. If R
R
d| x | 2 | f | 2 p dx R
R
d| f | 2 p dx γ = d (p d+2 − 1) −p σ (d−
∗M
∗γ2)
−1, σ ∗ (p) :=
4 d+2 (p− − 1) p
2(d (p+1) − 2)
d−p4(dp−4)for any f ∈ L p+1 ∩ D 1,2 ( R d ), then we have Z
R
d|∇ f | 2 dx+
Z
R
d| f | p+1 dx − K p,d
Z
R
d| f | 2 p dx γ
≥ C p,d R (p) [f ] 2 R
R
d| f | 2p dx γ
An improved Gagliardo-Nirenberg inequality (2/2)
A Csisz´ ar-Kullback inequality
R (p) [f ] ≥ C CK k f k 2 L p
2p(γ ( S −
d) 2) inf
g ∈ M
(p)d
k| f | 2 p − g 2p k 2 L
1( S
d)
with C CK = p− p+1 1 d+2 − 32 p p (d − 2) σ d
p−1 4p
∗ M ∗ 1 − γ . Let C p,d := C d,p C CK 2
Corollary
Under previous assumptions, we have Z
R
d|∇ f | 2 dx + Z
R
d| f | p+1 dx − K p,d
Z
R
d| f | 2p dx γ
≥ C p,d k f k 2p L
2p(γ ( S −
d) 4) inf
g ∈ M
d(p) k| f | 2 p − g 2 p k 4 L
1( S
d)
Conclusion 1: improved inequalities
We have found an improvement of an optimal Gagliardo-Nirenberg inequality, which provides an explicit measure of the distance to the manifold of optimal functions.
The method is based on the nonlinear flow
The explicit improvement gives (is equivalent to) an improved
entropy – entropy production inequality
Conclusion 2: improved rates
If m ∈ (m 1 , 1), with
f (t) := F σ(t) [u( · , t)]
σ(t ) = 1 K M
Z
R
d| x | 2 u(x, t) dx j(t) := J σ(t) [u( · , t)]
J σ [u] := m σ
d2(m − m
c) 1 − m
Z
R
du ∇ u m− 1 − ∇ B m− σ 1 2 dx we can write a system of coupled ODEs
f ′ = − j ≤ 0 σ ′ = − 2 d (1 m K − m)
2M
σ
d2(m−m
c) f ≤ 0 j ′ + 4 j = d 2 (m − m c ) h
j
σ + 4 d (1 − m) σ f i σ ′ − r
(3)
In the rescaled variables, we have found an improved decay (algebraic
rate) of the relative entropy. This is a new nonlinear effect, which
matters for the initial time layer
Conclusion 3: Best matching Barenblatt profiles are delayed
Let u be such that
v(τ, x) = µ d R(D τ) d u
1
2 log R(D τ), µ x R(D τ)
with τ 7→ R(τ) given as the solution to 1
R dR dτ =
µ 2 K M
Z
R
d| x | 2 v (τ, x) dx
−
d2(m−m
c)
, R(0) = 1 Then
1 R
dR dτ =
R 2 (τ) σ
1
2 log R(D τ)
−
d2(m − m
c)
that is R(τ) = R 0 (τ) ≤ R 0 (τ) where R 1 dR dτ
0= R 0 2 (τ ) σ (0) −
d2(m − m
c)
and asymptotically as τ → ∞ , R(τ) = R 0 (τ − δ) for some delay δ > 0
B. Fast diffusion equations:
New points of view
1
improved inequalities and scalings
2
scalings and a concavity property
3
improved rates and best matching
Improved inequalities and scalings
The logarithmic Sobolev inequality
d µ = µ dx, µ(x) = (2 π) −d/2 e −|x|
2/2 , on R d with d ≥ 1 Gaussian logarithmic Sobolev inequality
Z
R
2|∇ u | 2 dµ ≥ 1 2
Z
R
2| u | 2 log | u | 2 dµ for any function u ∈ H 1 ( R d , dµ) such that R
R
2| u | 2 dµ = 1 ϕ(t ) := d
4
exp 2 t
d
− 1 − 2 t d
∀ t ∈ R
[Bakry, Ledoux (2006)], [Fathi et al. (2014)], [Dolbeault, Toscani (2014)]
Proposition
Z
R
2|∇ u | 2 dµ − 1 2
Z
R
2| u | 2 log | u | 2 dµ ≥ ϕ Z
R
2| u | 2 log | u | 2 dµ
∀ u ∈ H 1 ( R d , d µ) s.t.
Z
R
2| u | 2 d µ = 1 and Z
R
2| x | 2 | u | 2 dµ = d
Consequences for the heat equation
Ornstein-Uhlenbeck equation (or backward Kolmogorov equation)
∂f
∂t = ∆f − x · ∇ f
with initial datum f 0 ∈ L 1 + ( R d , (1 + | x | 2 ) d µ and define the entropy as E [f ] :=
Z
R
2f log f d µ , d
dt E [f ] = − 4 Z
R
2|∇ √
f | 2 dµ ≤ − 2 E [f ] thus proving that E [f (t, · )] ≤ E [f 0 ] e − 2t . Moreover,
d dt
Z
R
2f | x | 2 dµ = 2 Z
R
2f (d − | x | 2 ) dµ Theorem
Assume that E [f 0 ] is finite and R
R
2f 0 | x | 2 dµ = d R
R
2f 0 dµ. Then E [f (t , · )] ≤ − d
2 log h 1 −
1 − e −
2dE [f
0] e − 2t i
∀ t ≥ 0
Gagliardo-Nirenberg inequalities and the FDE
k∇ w k ϑ L
2( S
d) k w k 1 L −
q+1ϑ ( S
d) ≥ C GN k w k L
2q( S
d)
With the right choice of the constants, the functional J[w ] := 1
4 (q 2 − 1) Z
R
d|∇ w | 2 dx+β Z
R
d| w | q+1 dx −K C α GN
Z
R
d| w | 2q dx
2qαis nonnegative and J[w ] ≥ J[w ∗ ] = 0 Theorem
[Dolbeault-Toscani] For some nonnegative, convex, increasing ϕ J[w ] ≥ ϕ
β R
R
d| w ∗ | q+1 dx − R
R
d| w | q+1 dx for any w ∈ L q+1 ( R d ) such that R
R
d|∇ w | 2 dx < ∞ and R
R
d| w | 2q | x | 2 dx = R
R
dw ∗ 2q | x | 2 dx
Consequence for decay rates of relative R´enyi entropies:
see [Carrillo-Toscani]
Scalings and a concavity property
The fast diffusion equation in original variables
Consider the nonlinear diffusion equation in R d , d ≥ 1
∂u
∂t = ∆u p
with initial datum u(x, t = 0) = u 0 (x) ≥ 0 such that R
R
du 0 dx = 1 and R
R
d| x | 2 u 0 dx < + ∞ . The large time behavior of the solutions is governed by the source-type Barenblatt solutions
U ⋆ (t, x) := 1
κ t 1/µ d B ⋆ x κ t 1/µ
where
µ := 2 + d (p − 1) , κ := 2 µ p p − 1
1/µ
and B ⋆ is the Barenblatt profile B ⋆ (x) :=
C ⋆ − | x | 2 1/(p − 1)
+ if p > 1 C ⋆ + | x | 2 1/(p− 1)
if p < 1
The entropy
The entropy is defined by E :=
Z
R
du p dx and the Fisher information by
I :=
Z
R
du |∇ v | 2 dx with v = p p − 1 u p − 1 If u solves the fast diffusion equation, then
E ′ = (1 − p) I To compute I ′ , we will use the fact that
∂v
∂t = (p − 1) v ∆v + |∇ v | 2 F := E σ with σ = µ
d (1 − p) = 1+ 2 1 − p
1
d + p − 1
= 2 d
1
1 − p − 1
has a linear growth asymptotically as t → + ∞
The concavity property
Theorem
[Toscani-Savar´e] Assume that p ≥ 1 − 1 d if d > 1 and p > 0 if d = 1.
Then F (t) is increasing, (1 − p) F ′′ (t ) ≤ 0 and
t→ lim + ∞
1
t F(t) = (1 − p) σ lim
t→ + ∞ E σ − 1 I = (1 − p) σ E σ ⋆ − 1 I ⋆
[Dolbeault-Toscani] The inequality
E σ − 1 I ≥ E σ ⋆ − 1 I ⋆
is equivalent to the Gagliardo-Nirenberg inequality k∇ w k θ L
2( S
d) k w k 1 L −
q+1θ ( S
d) ≥ C GN k w k L
2q( S
d)
if 1 − 1 d ≤ p < 1. Hint: u p − 1/2 = kwk w
L2q(Sd)
, q = 2 p− 1 1
The proof
Lemma
I ′ = d dt
Z
R
du |∇ v | 2 dx = − 2 Z
R
du p
k D 2 v k 2 + (p − 1) (∆v) 2 dx
k D 2 v k 2 = 1
d (∆v) 2 +
D 2 v − 1 d ∆v Id
2
1
σ (1 − p) E 2 − σ (E σ ) ′′ = (1 − p) (σ − 1) Z
R
du |∇ v | 2 dx 2
− 2 1
d + p − 1 Z
R
du p dx Z
R
du p (∆v ) 2 dx
− 2 Z
R
du p dx Z
R
du p
D 2 v − 1 d ∆v Id
2
dx
Improved rates and best matching
Temperature (fast diffusion case)
The second moment functional (temperature) is defined by Θ(t ) := 1
d Z
R
d| x | 2 u(t, x) dx and such that
Θ ′ = 2 E
0 t
τ(0)
τ(t) τ
∞= lim
s→+∞τ (s) s Θ
µ/2(s) s (s τ(0))Θ
s (s
+τ(t))Θ
µ/2µ/2
s (s
+τ
∞)Θ
µ/2s
Fast diffusion case
+Temperature (porous medium case) and delay
Let U ⋆ s be the best matching Barenblatt function, in the sense of relative entropy F [u | U ⋆ s ], among all Barenblatt functions ( U ⋆ s ) s>0 . We define s as a function of t and consider the delay given by
τ(t ) :=
Θ(t ) Θ ⋆
µ2− t
0
s τ(t) τ(0)
τ
∞= lim
s→+∞τ(s) t
s (s
+τ
(0)) Θ
µ/2s Θ
µ/2(s)
s (s
+τ (t))
Θ
µ/2s (s
+τ
∞) Θ
µ/2Porous medium case
A result on delays
Theorem
Assume that p ≥ 1 − 1 d and p 6 = 1. The best matching Barenblatt function of a solution u is (t, x) 7→ U ⋆ (t + τ(t), x) and the function t 7→ τ(t ) is nondecreasing if p > 1 and nonincreasing if 1 − d 1 ≤ p < 1 With G := Θ 1 −
η2, η = d (1 − p) = 2 − µ, the R´enyi entropy power functional H := Θ −
η2E is such that
G ′ = µ H with H := Θ −
η2E H ′
1 − p = Θ − 1 −
η2Θ I − d E 2
= d E 2
Θ
η2+1 (q − 1) with q := Θ I d E 2 ≥ 1 d E 2 = 1
d
− Z
R
dx · ∇ (u p ) dx 2
= 1 d
Z
R
dx · u ∇ v dx 2
≤ 1 d
Z
R
du | x | 2 dx Z
R
du |∇ v | 2 dx = Θ I
An estimate of the delay
Theorem
If p > 1 − 1 d and p 6 = 1, then the delay satisfies
t → lim + ∞ | τ(t ) − τ(0) | ≥ | 1 − p | Θ(0) 1 −
d2(1 −p) 2 H ⋆
H ⋆ − H(0) 2
Θ(0) I(0) − d E(0) 2
C. Fast diffusion equations on manifolds and sharp functional
inequalities
1
The sphere
2
The line
3
Compact Riemannian manifolds
4
The cylinder: Caffarelli-Kohn-Nirenberg
inequalities
Interpolation inequalities on the sphere
Joint work with M.J. Esteban, M. Kowalczyk and M. Loss
A family of interpolation inequalities on the sphere
The following interpolation inequality holds on the sphere p − 2
d Z
S
d|∇ u | 2 d v g + Z
S
d| u | 2 d v g ≥ Z
S
d| u | p d v g
2/p
∀ u ∈ H 1 ( S d , dv g ) for any p ∈ (2, 2 ∗ ] with 2 ∗ = d 2d − 2 if d ≥ 3
for any p ∈ (2, ∞ ) if d = 2
Here dv g is the uniform probability measure: v g ( S d ) = 1
1 is the optimal constant, equality achieved by constants
p = 2 ∗ corresponds to Sobolev’s inequality...
Stereographic projection
Sobolev inequality
The stereographic projection of S d ⊂ R d × R ∋ (ρ φ, z) onto R d : to ρ 2 + z 2 = 1, z ∈ [ − 1, 1], ρ ≥ 0, φ ∈ S d− 1 we associate x ∈ R d such that r = | x | , φ = | x x |
z = r 2 − 1
r 2 + 1 = 1 − 2
r 2 + 1 , ρ = 2 r r 2 + 1
and transform any function u on S d into a function v on R d using u(y ) = ρ r
d−22v(x) = r
22 +1
d−22v(x) = (1 − z ) −
d−22v(x) p = 2 ∗ , S d = 1 4 d (d − 2) | S d | 2/d : Euclidean Sobolev inequality
Z
R
d|∇ v | 2 dx ≥ S d
Z
R
d| v |
d2−2ddx
d−d2∀ v ∈ D 1,2 ( R d )
Extended inequality
Z
S
d|∇ u | 2 d v g ≥ d p − 2
"Z
S
d| u | p d v g
2/p
− Z
S
d| u | 2 d v g
#
∀ u ∈ H 1 ( S d , dµ) is valid
for any p ∈ (1, 2) ∪ (2, ∞ ) if d = 1, 2 for any p ∈ (1, 2) ∪ (2, 2 ∗ ] if d ≥ 3 Case p = 2: Logarithmic Sobolev inequality Z
S
d|∇ u | 2 d v g ≥ d 2
Z
S
d| u | 2 log
R | u | 2
S
d| u | 2 d v g
d v g ∀ u ∈ H 1 ( S d , dµ) Case p = 1: Poincar´e inequality
Z
S
d|∇ u | 2 d v g ≥ d Z
S
d| u − u ¯ | 2 d v g with u ¯ :=
Z
S
du d v g ∀ u ∈ H 1 ( S d , dµ)
Optimality: a perturbation argument
For any p ∈ (1, 2 ∗ ] if d ≥ 3, any p > 1 if d = 1 or 2, it is remarkable that
Q [u] := (p − 2) k∇ u k 2 L
2( S
d)
k u k 2 L
p( S
d) − k u k 2 L
2( S
d)
≥ inf
u ∈ H
1( S
d,dµ) Q [u] = 1 d is achieved in the limiting case
Q [1 + ε v] ∼ k∇ v k 2 L
2( S
d)
k v k 2 L
2( S
d)
as ε → 0 when v is an eigenfunction associated with the first nonzero eigenvalue of ∆ g , thus proving the optimality
p < 2: a proof by semi-groups using Nelson’s hypercontractivity lemma. p > 2: no simple proof based on spectral analysis is available:
[Beckner], an approach based on Lieb’s duality, the Funk-Hecke formula and some (non-trivial) computations
elliptic methods / Γ 2 formalism of Bakry-Emery / nonlinear flows
Schwarz symmetrization and the ultraspherical setting
(ξ 0 , ξ 1 , . . . ξ d ) ∈ S d , ξ d = z, P d
i=0 | ξ i | 2 = 1 [Smets-Willem]
Lemma
Up to a rotation, any minimizer of Q depends only on ξ d = z
• Let dσ(θ) := (sin Z θ)
dd−1d θ, Z d := √ π Γ(
d 2 ) Γ( d+1 2 )
: ∀ v ∈ H 1 ([0, π], dσ) p − 2
d Z π
0 | v ′ (θ) | 2 dσ + Z π
0 | v (θ) | 2 d σ ≥ Z π
0 | v (θ) | p dσ
2p• Change of variables z = cos θ, v(θ) = f (z ) p − 2
d Z 1
− 1 | f ′ | 2 ν dν d + Z 1
− 1 | f | 2 dν d ≥ Z 1
− 1 | f | p dν d
2pwhere ν d (z ) dz = dν d (z) := Z d − 1 ν
d2− 1 dz , ν(z ) := 1 − z 2
The ultraspherical operator
With dν d = Z d − 1 ν
d2− 1 dz, ν(z) := 1 − z 2 , consider the space L 2 (( − 1, 1), dν d ) with scalar product
h f 1 , f 2 i = Z 1
− 1
f 1 f 2 dν d , k f k p = Z 1
− 1
f p dν d
p1The self-adjoint ultraspherical operator is
L f := (1 − z 2 ) f ′′ − d z f ′ = ν f ′′ + d 2 ν ′ f ′ which satisfies h f 1 , L f 2 i = − R 1
− 1 f 1 ′ f 2 ′ ν dν d
Proposition
Let p ∈ [1, 2) ∪ (2, 2 ∗ ], d ≥ 1
−h f , L f i = Z 1
− 1 | f ′ | 2 ν dν d ≥ d k f k 2 p − k f k 2 2
p − 2 ∀ f ∈ H 1 ([ − 1, 1], d ν d )
Flows on the sphere
Heat flow and the Bakry-Emery method
Fast diffusion (porous media) flow and the choice of the exponents
Joint work with M.J. Esteban, M. Kowalczyk and M. Loss
Heat flow and the Bakry-Emery method
With g = f p , i.e. f = g α with α = 1/p
(Ineq.) −h f , L f i = −h g α , L g α i =: I [g ] ≥ d k g k 2 1 α − k g 2 α k 1
p − 2 =: F [g ] Heat flow
∂g
∂t = L g d
dt k g k 1 = 0 , d
dt k g 2 α k 1 = − 2 (p − 2) h f , L f i = 2 (p − 2) Z 1
− 1 | f ′ | 2 ν dν d
which finally gives d
dt F [g (t , · )] = − d p − 2
d
dt k g 2 α k 1 = − 2 d I [g (t, · )]
Ineq. ⇐⇒ d
dt F [g (t , · )] ≤ − 2 d F [g (t, · )] ⇐ = d
dt I [g (t, · )] ≤ − 2 d I [g (t , · )]
The equation for g = f p can be rewritten in terms of f as
∂f
∂t = L f + (p − 1) | f ′ | 2 f ν
− 1 2
d dt
Z 1
− 1 | f ′ | 2 ν dν d = 1 2
d
dt h f , L f i = hL f , L f i + (p − 1) h | f ′ | 2 f ν, L f i d
dt I [g (t, · )] + 2 d I [g (t, · )] = d dt
Z 1
− 1 | f ′ | 2 ν dν d + 2 d Z 1
− 1 | f ′ | 2 ν dν d
= − 2 Z 1
− 1
| f ′′ | 2 + (p − 1) d d + 2
| f ′ | 4
f 2 − 2 (p − 1) d − 1 d + 2
| f ′ | 2 f ′′
f
ν 2 dν d
is nonpositive if
| f ′′ | 2 + (p − 1) d d + 2
| f ′ | 4
f 2 − 2 (p − 1) d − 1 d + 2
| f ′ | 2 f ′′
f is pointwise nonnegative, which is granted if
(p − 1) d − 1 d + 2
2
≤ (p − 1) d
d + 2 ⇐⇒ p ≤ 2 d 2 + 1
(d − 1) 2 = 2 # < 2 d
d − 2 = 2 ∗
... up to the critical exponent: a proof in two slides
d dz , L
u = ( L u) ′ − L u ′ = − 2 z u ′′ − d u ′ Z 1
− 1
( L u) 2 dν d = Z 1
− 1 | u ′′ | 2 ν 2 d ν d + d Z 1
− 1 | u ′ | 2 ν dν d
Z 1
− 1
( L u) | u ′ | 2
u ν dν d = d d + 2
Z 1
− 1
| u ′ | 4
u 2 ν 2 dν d − 2 d − 1 d + 2
Z 1
− 1
| u ′ | 2 u ′′
u ν 2 dν d
On ( − 1, 1), let us consider the porous medium (fast diffusion) flow u t = u 2 − 2β
L u + κ | u ′ | 2 u ν
If κ = β (p − 2) + 1, the L p norm is conserved d
dt Z 1
− 1
u βp dν d = β p (κ − β (p − 2) − 1) Z 1
− 1
u β(p − 2) | u ′ | 2 ν dν d = 0
f = u β , k f ′ k 2 L
2( S
d) + p − d 2
k f k 2 L
2( S
d) − k f k 2 L
p( S
d)
≥ 0 ?
A :=
Z 1
− 1 | u ′′ | 2 ν 2 dν d − 2 d − 1
d + 2 (κ + β − 1) Z 1
− 1
u ′′ | u ′ | 2 u ν 2 dν d
+
κ (β − 1) + d
d + 2 (κ + β − 1) Z 1
− 1
| u ′ | 4 u 2 ν 2 dν d
A is nonnegative for some β if 8 d 2
(d + 2) 2 (p − 1) (2 ∗ − p) ≥ 0
A is a sum of squares if p ∈ (2, 2 ∗ ) for an arbitrary choice of β in a certain interval (depending on p and
A = Z 1
− 1
u ′′ − p + 2 6 − p
| u ′ | 2 u
2
ν 2 dν d ≥ 0 if p = 2 ∗ and β = 4
6 − p
The rigidity point of view
Which computation have we done ? u t = u 2 − 2β
L u + κ |u
′
|
2u ν
− L u − (β − 1) | u ′ | 2
u ν + λ
p − 2 u = λ p − 2 u κ Multiply by L u and integrate
...
Z 1
− 1 L u u κ dν d = − κ Z 1
− 1
u κ | u ′ | 2 u dν d
Multiply by κ | u
′
|
2u and integrate ... = + κ
Z 1
− 1
u κ | u ′ | 2 u dν d
The two terms cancel and we are left only with the two-homogenous
terms
Improvements of the inequalities (subcritical range)
as long as the exponent is either in the range (1, 2) or in the range (2, 2 ∗ ), on can establish improved inequalities
An improvement automatically gives an explicit stability result of the optimal functions in the (non-improved) inequality
By duality, this provides a stability result for Keller-Lieb-Tirring
inequalities
What does “improvement” mean ?
An improved inequality is
d Φ (e) ≤ i ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L
2( S
d) = 1 for some function Φ such that Φ(0) = 0, Φ ′ (0) = 1, Φ ′ > 0 and Φ(s) > s for any s. With Ψ(s) := s − Φ − 1 (s)
i − d e ≥ d (Ψ ◦ Φ)(e) ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L
2( S
d) = 1 Lemma (Generalized Csisz´ar-Kullback inequalities)
k∇ u k 2 L
2( S
d) − d p − 2
h k u k 2 L
p( S
d) − k u k 2 L
2( S
d)
i
≥ d k u k 2 L
2( S
d) (Ψ ◦ Φ)
C k u k
2 (1−r) Ls(Sd)